Theorem 2sqlem11 25154

Description: Lemma for 2sq 25155. (Contributed by Mario Carneiro, 19-Jun-2015.)

Hypotheses

Ref	Expression
2sq.1	|- S = ran ( w e. ZZ[_i] |-> ( ( abs ` w ) ^ 2 ) )
2sqlem7.2	|- Y = { z | E. x e. ZZ E. y e. ZZ ( ( x gcd y ) = 1 /\ z = ( ( x ^ 2 ) + ( y ^ 2 ) ) ) }

Assertion

Ref	Expression
2sqlem11	|- ( ( P e. Prime /\ ( P mod 4 ) = 1 ) -> P e. S )

Distinct variable groups:   x, w, y, z   x, S, y, z   x, Y, y   x, P, y

Allowed substitution hints:   P( z, w)   S( w)   Y( z, w)

Proof of Theorem 2sqlem11

Dummy variable n is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpr 477	. . . . 5 |- ( ( P e. Prime /\ ( P mod 4 ) = 1 ) -> ( P mod 4 ) = 1 )
2		simpl 473	. . . . . . 7 |- ( ( P e. Prime /\ ( P mod 4 ) = 1 ) -> P e. Prime )
3		1ne2 11240	. . . . . . . . . . 11 |- 1 =/= 2
4	3	necomi 2848	. . . . . . . . . 10 |- 2 =/= 1
5		oveq1 6657	. . . . . . . . . . . 12 |- ( P = 2 -> ( P mod 4 ) = ( 2 mod 4 ) )
6		2re 11090	. . . . . . . . . . . . 13 |- 2 e. RR
7		4re 11097	. . . . . . . . . . . . . 14 |- 4 e. RR
8		4pos 11116	. . . . . . . . . . . . . 14 |- 0 < 4
9	7, 8	elrpii 11835	. . . . . . . . . . . . 13 |- 4 e. RR+
10		0le2 11111	. . . . . . . . . . . . 13 |- 0 <_ 2
11		2lt4 11198	. . . . . . . . . . . . 13 |- 2 < 4
12		modid 12695	. . . . . . . . . . . . 13 |- ( ( ( 2 e. RR /\ 4 e. RR+ ) /\ ( 0 <_ 2 /\ 2 < 4 ) ) -> ( 2 mod 4 ) = 2 )
13	6, 9, 10, 11, 12	mp4an 709	. . . . . . . . . . . 12 |- ( 2 mod 4 ) = 2
14	5, 13	syl6eq 2672	. . . . . . . . . . 11 |- ( P = 2 -> ( P mod 4 ) = 2 )
15	14	neeq1d 2853	. . . . . . . . . 10 |- ( P = 2 -> ( ( P mod 4 ) =/= 1 <-> 2 =/= 1 ) )
16	4, 15	mpbiri 248	. . . . . . . . 9 |- ( P = 2 -> ( P mod 4 ) =/= 1 )
17	16	necon2i 2828	. . . . . . . 8 |- ( ( P mod 4 ) = 1 -> P =/= 2 )
18	1, 17	syl 17	. . . . . . 7 |- ( ( P e. Prime /\ ( P mod 4 ) = 1 ) -> P =/= 2 )
19		eldifsn 4317	. . . . . . 7 |- ( P e. ( Prime \ { 2 } ) <-> ( P e. Prime /\ P =/= 2 ) )
20	2, 18, 19	sylanbrc 698	. . . . . 6 |- ( ( P e. Prime /\ ( P mod 4 ) = 1 ) -> P e. ( Prime \ { 2 } ) )
21		m1lgs 25113	. . . . . 6 |- ( P e. ( Prime \ { 2 } ) -> ( ( -u 1 /L P ) = 1 <-> ( P mod 4 ) = 1 ) )
22	20, 21	syl 17	. . . . 5 |- ( ( P e. Prime /\ ( P mod 4 ) = 1 ) -> ( ( -u 1 /L P ) = 1 <-> ( P mod 4 ) = 1 ) )
23	1, 22	mpbird 247	. . . 4 |- ( ( P e. Prime /\ ( P mod 4 ) = 1 ) -> ( -u 1 /L P ) = 1 )
24		neg1z 11413	. . . . 5 |- -u 1 e. ZZ
25		lgsqr 25076	. . . . 5 |- ( ( -u 1 e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> ( ( -u 1 /L P ) = 1 <-> ( -. P || -u 1 /\ E. n e. ZZ P || ( ( n ^ 2 ) - -u 1 ) ) ) )
26	24, 20, 25	sylancr 695	. . . 4 |- ( ( P e. Prime /\ ( P mod 4 ) = 1 ) -> ( ( -u 1 /L P ) = 1 <-> ( -. P || -u 1 /\ E. n e. ZZ P || ( ( n ^ 2 ) - -u 1 ) ) ) )
27	23, 26	mpbid 222	. . 3 |- ( ( P e. Prime /\ ( P mod 4 ) = 1 ) -> ( -. P || -u 1 /\ E. n e. ZZ P || ( ( n ^ 2 ) - -u 1 ) ) )
28	27	simprd 479	. 2 |- ( ( P e. Prime /\ ( P mod 4 ) = 1 ) -> E. n e. ZZ P || ( ( n ^ 2 ) - -u 1 ) )
29		simprl 794	. . . . 5 |- ( ( ( P e. Prime /\ ( P mod 4 ) = 1 ) /\ ( n e. ZZ /\ P || ( ( n ^ 2 ) - -u 1 ) ) ) -> n e. ZZ )
30		1zzd 11408	. . . . 5 |- ( ( ( P e. Prime /\ ( P mod 4 ) = 1 ) /\ ( n e. ZZ /\ P || ( ( n ^ 2 ) - -u 1 ) ) ) -> 1 e. ZZ )
31		gcd1 15249	. . . . . 6 |- ( n e. ZZ -> ( n gcd 1 ) = 1 )
32	31	ad2antrl 764	. . . . 5 |- ( ( ( P e. Prime /\ ( P mod 4 ) = 1 ) /\ ( n e. ZZ /\ P || ( ( n ^ 2 ) - -u 1 ) ) ) -> ( n gcd 1 ) = 1 )
33		eqidd 2623	. . . . 5 |- ( ( ( P e. Prime /\ ( P mod 4 ) = 1 ) /\ ( n e. ZZ /\ P || ( ( n ^ 2 ) - -u 1 ) ) ) -> ( ( n ^ 2 ) + 1 ) = ( ( n ^ 2 ) + 1 ) )
34		oveq1 6657	. . . . . . . 8 |- ( x = n -> ( x gcd y ) = ( n gcd y ) )
35	34	eqeq1d 2624	. . . . . . 7 |- ( x = n -> ( ( x gcd y ) = 1 <-> ( n gcd y ) = 1 ) )
36		oveq1 6657	. . . . . . . . 9 |- ( x = n -> ( x ^ 2 ) = ( n ^ 2 ) )
37	36	oveq1d 6665	. . . . . . . 8 |- ( x = n -> ( ( x ^ 2 ) + ( y ^ 2 ) ) = ( ( n ^ 2 ) + ( y ^ 2 ) ) )
38	37	eqeq2d 2632	. . . . . . 7 |- ( x = n -> ( ( ( n ^ 2 ) + 1 ) = ( ( x ^ 2 ) + ( y ^ 2 ) ) <-> ( ( n ^ 2 ) + 1 ) = ( ( n ^ 2 ) + ( y ^ 2 ) ) ) )
39	35, 38	anbi12d 747	. . . . . 6 |- ( x = n -> ( ( ( x gcd y ) = 1 /\ ( ( n ^ 2 ) + 1 ) = ( ( x ^ 2 ) + ( y ^ 2 ) ) ) <-> ( ( n gcd y ) = 1 /\ ( ( n ^ 2 ) + 1 ) = ( ( n ^ 2 ) + ( y ^ 2 ) ) ) ) )
40		oveq2 6658	. . . . . . . 8 |- ( y = 1 -> ( n gcd y ) = ( n gcd 1 ) )
41	40	eqeq1d 2624	. . . . . . 7 |- ( y = 1 -> ( ( n gcd y ) = 1 <-> ( n gcd 1 ) = 1 ) )
42		oveq1 6657	. . . . . . . . . 10 |- ( y = 1 -> ( y ^ 2 ) = ( 1 ^ 2 ) )
43		sq1 12958	. . . . . . . . . 10 |- ( 1 ^ 2 ) = 1
44	42, 43	syl6eq 2672	. . . . . . . . 9 |- ( y = 1 -> ( y ^ 2 ) = 1 )
45	44	oveq2d 6666	. . . . . . . 8 |- ( y = 1 -> ( ( n ^ 2 ) + ( y ^ 2 ) ) = ( ( n ^ 2 ) + 1 ) )
46	45	eqeq2d 2632	. . . . . . 7 |- ( y = 1 -> ( ( ( n ^ 2 ) + 1 ) = ( ( n ^ 2 ) + ( y ^ 2 ) ) <-> ( ( n ^ 2 ) + 1 ) = ( ( n ^ 2 ) + 1 ) ) )
47	41, 46	anbi12d 747	. . . . . 6 |- ( y = 1 -> ( ( ( n gcd y ) = 1 /\ ( ( n ^ 2 ) + 1 ) = ( ( n ^ 2 ) + ( y ^ 2 ) ) ) <-> ( ( n gcd 1 ) = 1 /\ ( ( n ^ 2 ) + 1 ) = ( ( n ^ 2 ) + 1 ) ) ) )
48	39, 47	rspc2ev 3324	. . . . 5 |- ( ( n e. ZZ /\ 1 e. ZZ /\ ( ( n gcd 1 ) = 1 /\ ( ( n ^ 2 ) + 1 ) = ( ( n ^ 2 ) + 1 ) ) ) -> E. x e. ZZ E. y e. ZZ ( ( x gcd y ) = 1 /\ ( ( n ^ 2 ) + 1 ) = ( ( x ^ 2 ) + ( y ^ 2 ) ) ) )
49	29, 30, 32, 33, 48	syl112anc 1330	. . . 4 |- ( ( ( P e. Prime /\ ( P mod 4 ) = 1 ) /\ ( n e. ZZ /\ P || ( ( n ^ 2 ) - -u 1 ) ) ) -> E. x e. ZZ E. y e. ZZ ( ( x gcd y ) = 1 /\ ( ( n ^ 2 ) + 1 ) = ( ( x ^ 2 ) + ( y ^ 2 ) ) ) )
50		ovex 6678	. . . . 5 |- ( ( n ^ 2 ) + 1 ) e. _V
51		eqeq1 2626	. . . . . . 7 |- ( z = ( ( n ^ 2 ) + 1 ) -> ( z = ( ( x ^ 2 ) + ( y ^ 2 ) ) <-> ( ( n ^ 2 ) + 1 ) = ( ( x ^ 2 ) + ( y ^ 2 ) ) ) )
52	51	anbi2d 740	. . . . . 6 |- ( z = ( ( n ^ 2 ) + 1 ) -> ( ( ( x gcd y ) = 1 /\ z = ( ( x ^ 2 ) + ( y ^ 2 ) ) ) <-> ( ( x gcd y ) = 1 /\ ( ( n ^ 2 ) + 1 ) = ( ( x ^ 2 ) + ( y ^ 2 ) ) ) ) )
53	52	2rexbidv 3057	. . . . 5 |- ( z = ( ( n ^ 2 ) + 1 ) -> ( E. x e. ZZ E. y e. ZZ ( ( x gcd y ) = 1 /\ z = ( ( x ^ 2 ) + ( y ^ 2 ) ) ) <-> E. x e. ZZ E. y e. ZZ ( ( x gcd y ) = 1 /\ ( ( n ^ 2 ) + 1 ) = ( ( x ^ 2 ) + ( y ^ 2 ) ) ) ) )
54		2sqlem7.2	. . . . 5 |- Y = { z | E. x e. ZZ E. y e. ZZ ( ( x gcd y ) = 1 /\ z = ( ( x ^ 2 ) + ( y ^ 2 ) ) ) }
55	50, 53, 54	elab2 3354	. . . 4 |- ( ( ( n ^ 2 ) + 1 ) e. Y <-> E. x e. ZZ E. y e. ZZ ( ( x gcd y ) = 1 /\ ( ( n ^ 2 ) + 1 ) = ( ( x ^ 2 ) + ( y ^ 2 ) ) ) )
56	49, 55	sylibr 224	. . 3 |- ( ( ( P e. Prime /\ ( P mod 4 ) = 1 ) /\ ( n e. ZZ /\ P || ( ( n ^ 2 ) - -u 1 ) ) ) -> ( ( n ^ 2 ) + 1 ) e. Y )
57		prmnn 15388	. . . 4 |- ( P e. Prime -> P e. NN )
58	57	ad2antrr 762	. . 3 |- ( ( ( P e. Prime /\ ( P mod 4 ) = 1 ) /\ ( n e. ZZ /\ P || ( ( n ^ 2 ) - -u 1 ) ) ) -> P e. NN )
59		simprr 796	. . . 4 |- ( ( ( P e. Prime /\ ( P mod 4 ) = 1 ) /\ ( n e. ZZ /\ P || ( ( n ^ 2 ) - -u 1 ) ) ) -> P || ( ( n ^ 2 ) - -u 1 ) )
60	29	zcnd 11483	. . . . . 6 |- ( ( ( P e. Prime /\ ( P mod 4 ) = 1 ) /\ ( n e. ZZ /\ P || ( ( n ^ 2 ) - -u 1 ) ) ) -> n e. CC )
61	60	sqcld 13006	. . . . 5 |- ( ( ( P e. Prime /\ ( P mod 4 ) = 1 ) /\ ( n e. ZZ /\ P || ( ( n ^ 2 ) - -u 1 ) ) ) -> ( n ^ 2 ) e. CC )
62		ax-1cn 9994	. . . . 5 |- 1 e. CC
63		subneg 10330	. . . . 5 |- ( ( ( n ^ 2 ) e. CC /\ 1 e. CC ) -> ( ( n ^ 2 ) - -u 1 ) = ( ( n ^ 2 ) + 1 ) )
64	61, 62, 63	sylancl 694	. . . 4 |- ( ( ( P e. Prime /\ ( P mod 4 ) = 1 ) /\ ( n e. ZZ /\ P || ( ( n ^ 2 ) - -u 1 ) ) ) -> ( ( n ^ 2 ) - -u 1 ) = ( ( n ^ 2 ) + 1 ) )
65	59, 64	breqtrd 4679	. . 3 |- ( ( ( P e. Prime /\ ( P mod 4 ) = 1 ) /\ ( n e. ZZ /\ P || ( ( n ^ 2 ) - -u 1 ) ) ) -> P || ( ( n ^ 2 ) + 1 ) )
66		2sq.1	. . . 4 |- S = ran ( w e. ZZ[_i] |-> ( ( abs ` w ) ^ 2 ) )
67	66, 54	2sqlem10 25153	. . 3 |- ( ( ( ( n ^ 2 ) + 1 ) e. Y /\ P e. NN /\ P || ( ( n ^ 2 ) + 1 ) ) -> P e. S )
68	56, 58, 65, 67	syl3anc 1326	. 2 |- ( ( ( P e. Prime /\ ( P mod 4 ) = 1 ) /\ ( n e. ZZ /\ P || ( ( n ^ 2 ) - -u 1 ) ) ) -> P e. S )
69	28, 68	rexlimddv 3035	1 |- ( ( P e. Prime /\ ( P mod 4 ) = 1 ) -> P e. S )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   {cab 2608   =/= wne 2794   E.wrex 2913   \ cdif 3571   {csn 4177   class class class wbr 4653   |-> cmpt 4729   ran crn 5115   ` cfv 5888  (class class class)co 6650   CCcc 9934   RRcr 9935   0cc0 9936   1c1 9937   + caddc 9939   < clt 10074   <_ cle 10075   - cmin 10266   -ucneg 10267   NNcn 11020   2c2 11070   4c4 11072   ZZcz 11377   RR+crp 11832   mod cmo 12668   ^cexp 12860   abscabs 13974   || cdvds 14983   gcd cgcd 15216   Primecprime 15385   ZZ[_i]cgz 15633   /Lclgs 25019

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014  ax-addf 10015  ax-mulf 10016

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-ofr 6898  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-tpos 7352  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-ec 7744  df-qs 7748  df-map 7859  df-pm 7860  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-xnn0 11364  df-z 11378  df-dec 11494  df-uz 11688  df-q 11789  df-rp 11833  df-fz 12327  df-fzo 12466  df-fl 12593  df-mod 12669  df-seq 12802  df-exp 12861  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-dvds 14984  df-gcd 15217  df-prm 15386  df-phi 15471  df-pc 15542  df-gz 15634  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-starv 15956  df-sca 15957  df-vsca 15958  df-ip 15959  df-tset 15960  df-ple 15961  df-ds 15964  df-unif 15965  df-hom 15966  df-cco 15967  df-0g 16102  df-gsum 16103  df-prds 16108  df-pws 16110  df-imas 16168  df-qus 16169  df-mre 16246  df-mrc 16247  df-acs 16249  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-mhm 17335  df-submnd 17336  df-grp 17425  df-minusg 17426  df-sbg 17427  df-mulg 17541  df-subg 17591  df-nsg 17592  df-eqg 17593  df-ghm 17658  df-cntz 17750  df-cmn 18195  df-abl 18196  df-mgp 18490  df-ur 18502  df-srg 18506  df-ring 18549  df-cring 18550  df-oppr 18623  df-dvdsr 18641  df-unit 18642  df-invr 18672  df-dvr 18683  df-rnghom 18715  df-drng 18749  df-field 18750  df-subrg 18778  df-lmod 18865  df-lss 18933  df-lsp 18972  df-sra 19172  df-rgmod 19173  df-lidl 19174  df-rsp 19175  df-2idl 19232  df-nzr 19258  df-rlreg 19283  df-domn 19284  df-idom 19285  df-assa 19312  df-asp 19313  df-ascl 19314  df-psr 19356  df-mvr 19357  df-mpl 19358  df-opsr 19360  df-evls 19506  df-evl 19507  df-psr1 19550  df-vr1 19551  df-ply1 19552  df-coe1 19553  df-evl1 19681  df-cnfld 19747  df-zring 19819  df-zrh 19852  df-zn 19855  df-mdeg 23815  df-deg1 23816  df-mon1 23890  df-uc1p 23891  df-q1p 23892  df-r1p 23893  df-lgs 25020

This theorem is referenced by:  2sq  25155